Highlight the limits of a student’s existing skills and knowledge. New mathematical tools are often developed to account for the limitations of the old ones. You can’t model the path of a basketball with linear equations — we need quadratics. You can’t model the growth of bacteria with a quadratic equations — we need exponentials. Offer students a challenge for which their old skills look useful but turn out to be ineffective. That moment of cognitive conflict can engage students in a discussion of new tools and counter the perception that math is a disjointed set of rules and procedures, each bearing no relationship to the one preceding it.

I used as my leading question: How can we order fractions with different denominators?

We came the position that it would be much easier if they had the same denominator. Hey presto, this is where I stepped in with the idea of percentages – A way of comparing fractional amounts, where the denominator is always 100.

I didn’t work perfectly, as these thing rarely do, but I got through to some of them, and it’s something I can build on.